Tomas Hagel set up Kucherov for a tap-in on a 2-on-1.
Hedman opened the scoring 32 seconds into the game, and Karlsson tied it 50 seconds later on a blast from the point.
Colton gave Tampa Bay a 2-1 lead with 6:16 left in the first period. Sturm tied it with his fifth of the season at 5:36 of the second, and Killorn put the Lightning back in front with 1:19 left in the second.
Karlsson tied it again on a power play at 4:34 of the third. He has six goals.
San Jose coach David Quinn remains a victory shy of the century mark.
Sharks: Continue six-game homestand against Anaheim on Tuesday night.
Lightning: Begin a four-game homestand against Ottawa on Tuesday night.
AP NHL: https://apnews.com/hub/nhl and https://twitter.com/AP_Sports
Max Miller, The Associated Press